International Journal for Computational Methods in Engineering Science and Mechanics, 15:390–400, 2014
c Taylor & Francis Group, LLC
Copyright ISSN: 1550-2287 print / 1550-2295 online
DOI: 10.1080/15502287.2014.915253
Finite Element Simulation of Two-Dimensional Pulsatile
Blood Flow Through a Stenosed Artery in the Presence
of External Magnetic Field
Haleh Alimohamadi1 and Mohsen Imani2
1
Department of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran
Department of Electrical and Computer Engineering, College of Engineering, University of Tehran,
Tehran, Iran
2
This paper introduces the impact of external magnetic field
on blood flow patterns in a stenosis artery. Considering the fatty
deposited lump, arterial walls as porous media, and pulsatile inflow base on human-heart-beating rate closes our model to the
actual stenosis blood artery. In this study, by solving transient
fluid dynamic equations in coupled porous and free media, the velocity, temperature, and shear stress distribution along the lump
are investigated. The results show that applying 105 magnetic
field intensity (MnF ) creates two vortexes on the lumps’ edges
and 15X (16.6X) higher shear stress (temperature) in the stenosis
region.
Keywords
Blood flow, Biomagnetic fluid, Stenosis artery, Heat
transfer, Porous media, Magnetization and Lorentz
forces
1. INTRODUCTION
Currently, cardiovascular diseases, especially stenosis, are
very prevalent and take the lives of many people. This common
disease occurs by backlogging macromolecules on the arterial
walls and atherosclerotic plaques’ formation by the passage of
the time. These plaques block the normal path of blood flow,
make it narrower and, in critical conditions, close it completely.
Biological research shows that these macromolecules are mostly
made from fatty material and are vulnerable to high temperature
and shear stress, so dissolving them by increasing heat transfer
rate or exerting higher viscous forces through applying external
Address correspondence to Mohsen Imani, Room 401, School of
Electrical and Computer Engineering, University College of Engineering, University of Tehran, P.O. Box 14395-515, North Kargar St.,
Tehran, Iran. E-mail: [email protected]
Color versions of one or more of the figures in the article can be
found online at www.tandfonline.com/ucme.
magnetic field is a safe and new bioengineering-suggested remedy. Natural blood has an innate, slight magnetization feature
and in order to increase this characteristic, additional injected
ferro-nanoparticles have recently been used. These particles increase blood magnetization and are mostly organic solvents
which can act as solutes. The solution of these nanoparticles in
natural and industrial fluids have wide application in bioengineering [1, 2], micro-electrical and mechanical structure [3],
cardiovascular and articulate treatments [4, 5], and two-phase
flow [6]. During the last few decades, numerous investigations
have been done in this new field of biomechanics. Stangeby and
Ethier [7] worked out a coupling model of Navier-Stokes and
Brinkman equations in free and porous media. The dynamic
expression for heat and mass transfer in stenosis artery is introduced in [8]. In [9], the velocity and temperature distribution of
biofluid flow under the influence of an external magnetic field
are discussed. The latter article modeled the artery as a uniform tub with solid and impenetrable walls. In [10], different
models of macro blood flow were coupled with various areas
and types of stenosis. The time-dependent heat transfer of twodimensional blood flow is discussed in [11]; however, this article
neglected the porous assumption for arterial walls. The effects
of permeability, magnetic field, and body acceleration on blood
flow passing through a porous media are brought up in [12], and
the analytical solution of stationary non-Newtonian blood flow
in porous media at the present of magnetic field is investigated in
[13]. The innovation of this article lies in solving transient fluid
dynamics equations of blood flow through stenosis geometry
taking into account the non-Newtonian viscosity of blood and
both magnetization and Lorentz forces. In this study, according
to a real heart-beating rate, the time-dependent inlet velocity
alters and the impact of the magnetic field on different heart cycles is described. In Part 2, the assumed geometry for a stenosis
vessel in addition to the governing equations of fluid flow under
the action of magnetic field is discussed. Part 3 is developed to
390
391
PULSATILE BLOOD FLOW THROUGH A STENOSED ARTERY
TABLE 1
Constant value
Parameters
ρ
η
Re
T1
δT
Pr
Cp
μ0
η0
Numerical value
Parameters
Numerical value
1050 kg.m−3
3.2 × 10−3 kg.m−1 .s −1
0.04
300 0 K
30 0 K
24.95
14.65J.(Kg.K)−1
4π × 10−7
35 × 10−3 kg.m−1 .s −1
Ec
α
k
MnF
MnM
Da
χ0
β
n
8.7 × 10−6
1.22 × 10−7 m2 .s −1
1.832 × 10−3 J oule.0 K −1 m−1 s −1
[105:5×105]
150
40-400-4000
0.06
5.6×10−3 K −1
0.6
present the dimensionless equations and boundary conditions
and, finally, the simulation results and conclusion are outlined
in sections 4 and 5, respectively.
non-Newtonian flow under the action of the external magnetic
field are described as below:
Continuity equation:
∇.V ∗ = 0
2. ARTERY MODEL AND MATHEMATICAL
FORMULATION
2.1 Stenosis Artery Model
In this article, we consider a viscous, laminar, incompressible, transient and two-dimensional blood flow between two
porous plates. Uniform flow enters the domain and, in the middle of the vessel (Lf < x < Lf + Ls ), one porous lump on the
lower plate blocks the normal blood flow path. Both up and
down vessel walls are assumed at the constant temperature and
the length and height of the artery are shown by L and h, respectively. The flow is subjected to an external magnetic force
producing an infinite current plate along the z axis. The nonsymmetric stenosis blood artery is shown in Fig. 1 and presumed
as below function [14]:
(1)
Momentum equation [15]:
ρ
D V ∗
= −∇p∗ + η∇ 2 V ∗ + J ∗ × B ∗ + μ0 M ∗ ∇H ∗
Dt ∗
(2)
Energy equation [15]:
ρCp
∗
∗
DT ∗
J ∗ .J ∗
∗ ∂M DH
= k∇ 2 T ∗ + ηφ ∗ (3)
+
μ
T
−
0
Dt ∗
∂T ∗ Dt ∗
σ
where V ∗ = (u∗ , v ∗ )is the two-dimensional velocity field,
D
= ∂t∂∗ + V ∗ ∇ is the material derivative, ∇ = ( ∂x∂ ∗ , ∂y∂ ∗ )
Dt ∗
2
2
is the gradient operator, ∇ 2 = ∇.∇ = ( ∂x∂ ∗2 , ∂y∂ ∗2 ) is the Laplacian operator, ρ is the fluid density, P is the pressure, η is the
dynamic viscosity, μ0 is the magnetic permeability of vacuum,
M∗ is the magnetization, H∗ is the magnetic field intensity, B∗ is
the magnetic induction where B∗ = μ0 (M∗ + H∗ ), σ is the elec⎧
3d
3
2 2
0.7 − 2hL
4 [11(x − Lf )Ls − 47(x − Lf ) Ls
⎨
trical conductivity of the fluid, J∗ is the density of the electrical
y
= +72(x − Lf )3 Ls − 36(x − Lf )4 ]
Lf ≤ x ≤ Ls + Lf . current, T∗ is the temperature, k is the thermal conductivity, C
p
h ⎩
1
otherwise
is the specific heat at constant pressure, and ϕ ∗ is the dissipation
The porous assumption for arterial walls and the fatty de- factor. For the two-dimensional problem, it is written as:
∗ 2
∗ 2 ∗
posited plaque is conforming to actual behavior of our blood
∂u
∂v
∂v
∂u∗ 2
artery. As depicted in Fig. 1, 15% total artery thickness is asφ∗ = 2
+
2
+
+
(4)
∂x ∗
∂y ∗
∂x ∗
∂x ∗
signed to each porous arterial wall and “d” is the maximum
height of stenosis, which depends on the disease progression,
The μ0 M ∗ ∇H ∗ and J∗ ×B∗ terms in the momentum equation
and its increment can create a critical condition for patients.
demonstrate the magnetization and Lorentz forces, respectively,
which appear because of Magneto Hydro Dynamic (MHD) and
Ferro Hydro Dynamic (FHD) effects. In this article, the power
2.2 Heat Transfer and Fluid Flow Equations
law model is used for modeling non-Newtonian blood flow
Under the action of an external magnetic field, two kinds
viscosity [16]:
of forces affect the blood flow. One of them is magnetization
force, which appears due to magnetic field gradient, and the
(5)
η = η0 |γ̇ |n
second is the Lorentz force, which should be taken into account
because of the high electrical conductivity of blood flow and η0 and n are constant parameters and their values are cited in
ferronanoparticles’ solution. The governing equations on the Table 1.
392
H. ALIMOHAMADI AND M. IMANI
TABLE 2
Time variation of Nusselt number in different magnetic fields
with MnM = 102
Time Step
MnF
0
105
2×105
3×105
4×105
5×105
2
4
6
8
0.756
0.692
0.609
0.504
0.389
0.282
0.089
0.059
−0.006
−0.090
−0.179
−0.269
0.011
−0.012
−0.070
−0.147
−0.232
−0.317
−0.001
−0.023
−0.080
−0.155
−0.237
−0.322
2.3 The Brinkman Equations for the Porous Media Flow
As was mentioned before, the artery walls and the lump are
considered as porous media in this study. For simulating the
combination of porous and free flow, the Brinkman equations
have great accuracy. These equations for the porous regions are
expressed as:
Continuity equation:
∇.Vp∗ = 0
(6)
Momentum equation:
η
η ∗
ρ δ Vp∗
= −∇p∗ + ∇ 2 Vp∗ −
V + J ∗ × B ∗ + μ0 M ∗ ∇H ∗
εp δt ∗
εp
kbr p
(7)
−
→
where Vp∗ = (u∗p , vp∗ ),εp and kbr are velocity field, porosity,
and permeability of the porous regions. The magnetization and
Lorentz force are exerted to the porous regions as well. After obtaining the velocity field in the porous regions, the temperature
distribution in these domains is calculated by Eq. (3).
2.4 Magnetic Formula
For controlling blood flow in the artery, the magnetic field as
an external force is applied to the stenosis region. This magnetization is created by a current plate along the z axis and below
the stenosis region.
FIG. 1. Modeled stenosis artery.
According to Maxwell’s law, the external magnetic field is
explained based on the equations below:
∇ × H ∗ = J ∗ = σ (V ∗ × B ∗ )
∇.B ∗ = ∇.(H ∗ + M ∗ ) = 0
The magnetic field intensity components (Hx∗ , Hy∗ ) of a current plate are given by the expressions below:
∗
(x − x2∗ )2 + (y ∗ − y0∗ )2
H0
(10)
Ln
Hy∗ = −
2
(x ∗ − x1∗ )2 + (y ∗ − y0∗ )2
∗
∗
∗ (x − x2∗ )
−1 (x − x1 )
−
tan
(11)
Hx∗ = H0 tan−1
y ∗ − y0∗
y ∗ − y0∗
H ∗ = Hy∗ 2 + Hx ∗ 2
(12)
H0 is the magnetic field strength which depends on the applied magnetics induction (B∗ = μ0 (H∗ +M∗ )) and x∗ 1 , x∗ 2 , y∗ 0
are the positions of the horizontal plate, as shown in Fig. 2. The
magnetic field and magnetization force components in the entire 2D space of the artery are illustrated in Fig. 3. As described
earlier, magnetization force consists of multiplying magnetic
field and its gradient in different points of space. Thus, magnetic force is severely high at the edges of the plate where the
magnetic field starts and stops. This force affects blood flow
and diverts it from direct motion. The x-component of this force
attempts to stop the fluid motion and return it backward, while
the y-component drags ferroparticles upwards. These two forces
can produce circulating flow near the stenosis region, which is
the focus of this article.
TABLE 3
Effect of porosity value on maximum temperature with
different percent of artery stenosis and MnF = 105, MnM = 102
at time = 1
d/h
Darcy Number
40
400
4000
0.1
0.15
0.2
0.25
10.43
10.25
10.23
10.40
10.24
10.21
10.38
10.21
10.19
10.36
10.19
10.17
(8)
(9)
FIG. 2. The plate with current passes through the z direction.
393
PULSATILE BLOOD FLOW THROUGH A STENOSED ARTERY
2.5 Magnetization Equation
Magnetization property (M) is a feature of biofluid, which
determines the impression of magnetic field on the flow. Various
equations for magnetization property have been introduced; in
this article, the linear formula that relates the magnetization to
magnetic field strength and temperature is used [17]:
M ∗ = χm∗ H ∗
(13)
χm∗ is the magnetic susceptibility and varies with temperature:
χm∗ =
χ0
1 + β(T ∗ − T0 )
(14)
χ0 , βandT0 are constant parameters that are obtained by experimental data [17].
3. TRANSFORMATION OF EQUATIONS
In order to solve the coupled systems of equations, the following non-dimensional variables are introduced:
t =
α ∗
x∗
v∗
p∗
H∗
T∗
y∗
u∗
t x=
v=
p=
H =
T =
y=
u=
h2
h
h
ur
ur
ρu2r
H0
δT
(15)
where α = ρcK is the thermal diffusivity of fluid and ur = αh is
p
the characteristic velocity.
By substituting these non-dimensional variables into Eqs.
(1)–(7) and (14), we have:
In the free flow:
Continuity:
∂u ∂v
+
=0
∂x
∂y
(16)
x-momentum:
2
∂u
∂u
∂u
∂p
∂ u ∂ 2u
+u
+v
=−
+ Pr × |γ̇ |
+ 2
∂t
∂x
∂y
∂x
∂x 2
∂y
MnM
∂H
+
(χm + 1)2 (vHx Hy − uHy2 ) (17)
+MnF χm H
∂x
Re
y-momentum:
2
∂v
∂v
∂p
∂ v
∂v
∂ 2v
+u
+v
=−
+ Pr × |γ̇ |
+ 2
∂t
∂x
∂y
∂y
∂x 2
∂y
MnM
∂H
+
(uHx Hy − vHy2 )
+ MnF χm H
(18)
∂y
Re
FIG. 3. (a) Normalized total magnetic field intensity; (b) normalized x and y
magnetization force components produced by a fixed current plate.
Also, the energy equation is transformed into:
2
∂T
∂T
∂ T
∂T
∂ 2T
+ Ec × Pr × |γ̇ |
+u
+v
=
+
∂t
∂x
∂y
∂x 2
∂y 2
2 ∂u 2
∂v
∂v
∂u 2
+
+2
+
2
∂x
∂y
∂x
∂y
394
H. ALIMOHAMADI AND M. IMANI
FIG. 4. Stream function and temperature contours with velocity and temperature profiles at time = 1 in different locations.
MnM
∂χ
∂H
∂H
+
−MnF × Ec ×
T ×H u
+v
∂T
∂x
∂y
Re
×Ec × (χm + 1)2 × (uHy − vHx )2
(19)
For the porous region, the continuity and energy equations are similar to the free flow, but the momentum equation
is:
x momentum:
∂up
∂p
=−
+ Pr × |γ̇ |
∂t
∂x
2
∂ up
∂ 2 up
+ Pr × |γ̇ | × Da × up + MnF
+
∂x 2
∂y 2
MnM
∂H
+
(χm + 1)2 (vp Hx Hy − up Hy2 )
×χm × H
∂x
Re
(20)
PULSATILE BLOOD FLOW THROUGH A STENOSED ARTERY
395
The non-dimensional parameters which appear in the above
equations are:
Re =
Ec =
Pr =
Da =
MnF =
MnM =
FIG. 5. Shear stress in the stenosis region in different d/h at time = 1.
y momentum:
∂vp
∂vp
∂vp
∂p
+ up
+ vp
=−
+ Pr × |γ̇ |
∂t
∂x
∂y
∂y
2
∂ vp
∂ 2 vp
+ Pr × |γ̇ | × Da × vp + MnF
+
∂x 2
∂y 2
MnM
∂H
+
(uHx Hy − vHy2 )
χm H
∂y
Re
T0
δT
)
(28)
The two magnetic numbers appear because of magnetization
and Lorentz forces, and if they are set to zero, the problem reduces to an ordinary flow between two plates with heat transfer.
For the upper wall (y = 1 and 0 < x < 10), the temperature is constant (T1 ) and the velocity is zero (no
slip condition);
• For the lower plate (y = 0 and 0 < x < 10), the
temperature is constant (T1 ) and the velocity is zero
(no slip condition);
• For outlet (x = 10 and 0 < y < 1 ), the temperature
) and the pressure are zero;
gradient ( ∂T
∂x
•
(21)
χ0
1 + (βδT )(T −
μ20 H02 h2 σ
(Magnetic number of MHD)
η
3.1 Boundary Conditions
In order to solve Eqs. (16)–(21) simultaneously, the nondimensional boundary conditions below are applied:
where
χm =
ρα
hρur
=
(Reynolds number)
(23)
η
η
u2r
α2
=
(Eckert number)
(24)
Cp δT
Cp δT h2
η
(Prandtl number)
(25)
ρα
h2
(Darcy number)
(26)
kbr
μ0 H02 μ0 H02 h2
=
(Magnetic number of FHD)(27)
ρu2r
ρα 2
(22)
FIG. 6. Streamlines and temperature contours with MnF = 105 and MnM = 102 in different time steps.
396
H. ALIMOHAMADI AND M. IMANI
FIG. 7. The time variation of “u” velocity profile for different locations with MnF = 105 and MnM = 102.
For inlet (x = 0 and 0 < y < 1 ), the temperature is
constant (T1 ) and the uniform inflow velocity changes
with time according to [18];
• For the free-porous interface, it is assumed that velocity
−
→
is equal in these two regions (Vp = V .). This condition means that the velocity field remains continuous
between free and porous media.
•
4. SIMULATION RESULT
In this article, the current plate is fixed at x1 = 3.3, x2 = 6.7,
and y0 = −0.1 in order to encompass the entire critical region
and produce the maximum magnetic forces at the starting and
ending edges of the lump. The fluid is assumed to have nonNewtonian behavior and its electrical conducting is noticeable.
The time-dependent flow in the described geometry is solved
with COMSOL 4.3 code using a direct UMFPACK linear system solver. The constant values that are used in this paper are
archived in Table 1 [9, 11].
4.1 Flow in Stenosis Artery without Magnetic Force
In order to reach the fully developed flow before and after
the stenosis region, the ratio of stenosis length to total length
(Ls /L) is assumed to be small enough in this study. Thus, the
entry length is too small and the flow soon reaches the fully
developed condition, but in the stenosis region, due to a narrowing of the cross-sectional area, the velocity magnitude increases.
Fig. 4 shows the non-dimensional velocity and temperature contours with 20% artery stenosis. The dimensionless “u” velocity
component and temperature profiles in specific locations along
the vessel are shown in this figure. As expected, the flow has
fully developed behavior in each cross-section, and the velocity magnitude near the arterial walls (x = 2 graph in Fig. 4)
and inside the formed plaque (x = 4, 5, 6 graph in Fig. 4) is
extremely low because of the porosity feature of the walls and
lump.
In fluid mechanics, the temperature directly relates to the velocity magnitude and the fluid temperature increase if it moves
faster. As seen in Fig. 4, the stenosis region has the highest temperature because of stream line contraction and flow
acceleration in this area. As the added temperature graphs show,
the maximum increment of temperature in the stenosis region is
equal to 12mK (x = 6 in Fig. 4).
The dimensionless shear stress distribution on the lump is
elucidated in Fig. 5. By reaching to the first throat, the shear
stress value jumps severely due to increasing velocity gradient
above the lump. The shear stress decreases until the middle of
PULSATILE BLOOD FLOW THROUGH A STENOSED ARTERY
397
FIG. 8. The time variation of temperature profile in different x-positions with MnF = 105 and MnM = 102.
the lump region; after that, it rises up through the second throat.
As depicted in the figure, by increasing the stenosis percent, the
lump must tolerate higher shear stress because, with increasing
stenosis height, the available cross-sectional area for blood flow
becomes narrower and higher velocity gradients are exerted on
the lump. The shear stress trend and its increment, by increasing
the stenosis percent, coincides with calculated results in the
paper [14].
4.2 Time-Dependent Effect of Magnetic Field on Blood
Flow
The stream function and temperature contours at the present
external magnetic field and at three different time steps (t = 0.1,
0.3, 0.8) are demonstrated in Fig. 6. The primary effect of the
exerting magnetic field is the formation of two main vortexes
approximately at the edges of the current plate (x = 3.5 and
x = 6.6). The first vortex rotates counter-clockwise while the
second one revolves in the opposite direction. At t = 0.1, the
inlet velocity is negative, so the number of streamlines between
the two vortices are much fewer than at other times. With the
passage of time, the inlet velocity becomes positive and peaks
at t = 0.3 and t = 08; the power of viscosity and inertia forces
becomes remarkable and competes with magnetic forces. As
the figure at t = 0.8 shows, these two forces partly overcome
the magnetic force, increasing the number of streamlines in the
channel and weakening the vortices’ size and effect.
On the other hand, the temperature contours depict that, with
the passage of the time, not only does the maximum temperature go up, but it also focuses on the first and end edges of
the lump, where the vortices are created and the flow circulates. Atherosclerotic plaques are usually formed from fatty
deposits and can be dissolved in high ambient temperature. In
fact, the current plate acts as an external heat term that produces
two concentrated heat sources on the lump’s edges and can be
beneficial for eliminating these fatty plaques. Furthermore, the
formed vortices can be useful for targeting drug delivery since
the flow circulation creates stagnant flow above the specific area
and drug carriers could have more accessible time for activation
and absorption.
398
H. ALIMOHAMADI AND M. IMANI
FIG. 10. Shear stress distribution in the stenosis region with different values
of magnetic field intensity and MnM = 102 at time = 1.
is maximized and the fluid temperature is 0.2 K higher than the
walls.
FIG. 9. Temperature and v-velocity profile in the stenosis region with different
magnetic field intensity and MnM = 102 at time = 1.
The time variation of dimensionless axial velocity at the external magnetic field and for specific locations is depicted in Fig.
7. The location x = 2 is far from the magnetic plate and the effect
of the formed vortices in this region is less than in other areas. As
seen, with the passage of time the effect of the magnetic forces
declines because the inlet velocity increases and the viscous and
inertia forces become prominent such that the velocity profile
at t = 0.8 is similar to its parabolic profile without impact of
magnetization forces (see Fig. 4). The trend of velocity profiles
at x = 4 and x = 6 is reversed since the vortices in these regions
rotate in opposite directions. At x = 5, two opposite vortices neutralize each other’s impact and the “u” velocity profile in this
location resembles the figure with its non-magnetization effect
(see Fig. 4).
Increasing the temperature in respect to time and in different
locations is shown in Fig. 8. As the trend of this figure elucidates,
with the passage of time the fluid temperature increases such
that at x = 6, where the second vortex is formed, this increment
4.3 Effect of Magnetic Field Intensity on Fluid Flow
The magnetic field intensity has a direct impact on the penetration velocity, temperature, vortex size, and shear stress. The
effect of this parameter on the temperature and vertical velocity
along the stenosis region is demonstrated in Fig. 9. As shown in
this figure, by doubling and tripling the magnetic field intensity
from 105, the maximum velocity (temperature) increases about
2.25X and 3.4X (1.02X and 10.1X), respectively.
By strengthening the magnetic field, the bigger vortices influence a wider area and higher velocity gradient applied stronger
shear stress to the lump. By applying the magnetic field with
two (three) times stronger from 105, the maximum shear stress
will become 2X (2.8X) higher in the stenosis region.
Our simulation shows, at first time steps, that the blood temperature is lower than the arterial walls and heat transfers from
the walls to the fluid, but with time, the blood becomes warmer
and the heat transfer direction is reversed.
This occurrence is elucidated in Table 2, such that the Nusselt
number varies from a positive value to a negative value through
one heart cycle. The heat transfer rate speeds up by strengthening magnetic field intensity and bigger vortices formation,
and as the table shows, at t = 8 with a five-times increase in
magnetic field intensity from 105 to 5×105, the value of the
Nusselt number goes up about 13.5X. Generally, the results of
this section indicate that by applying a more powerful external
magnetic field, the magnitude of temperature, penetration velocity, and shear stress on the stenosis region go up remarkably
and the high value of these parameters can be useful for cutting,
corrosion, and dissolving the fatty lump from the vessel’s wall.
PULSATILE BLOOD FLOW THROUGH A STENOSED ARTERY
399
FIG. 11. Velocity arrows in the artery with different porosity values of walls and stenosis region with MnF = 105 and MnM = 102 at time = 1.
4.4 Effect of Porosity on Magneto-Therapy Performance
The porous permeability effect on the vortex power and penetration velocity in the stenosis area is illustrated in Fig. 11
and Fig. 12, respectively. Based on Fig. 11, a decreasing Darcy
number value causes bigger and stronger vortices to form in the
stenosis region, reducing the Darcy number about two orders of
magnitude and creating a six-times-higher vertical velocity and
penetration flow into the fatty lump (see Fig. 12). These results
make it clear that stenosis disease has better treatment under
a magneto-therapy process in the preliminary level because the
fatty lumps are more penetrable (lower Darcy number), and even
one weak external magnetic field can affect them.
Table 3 demonstrates the maximum blood temperature for
different porosity factors and stenosis heights. Increasing both
porosity and stenosis height parameters makes the magnetotherapy less efficient because, by increasing the value of the
Darcy number, the porous regions become more rigid, hardly resisting against the flow passage, and the available cross-sectional
area for blood flow declines. Thus, based on continuity law, the
velocity magnitude and, consequently, the inertia and viscosity
forces become powerful enough to attenuate the external magnetic field effect such that, by increasing the value of the Darcy
number about two orders of magnitude, the maximum blood
temperature declines at least 2%. In addition, stenosis progression has the same negative effect on the magneto-therapy effectiveness, as a lower temperature is produced on the fatty lump,
and obtaining higher temperature on the stenosis region can
help to dissolve the lump tissue faster and more easily. Hence,
according to this table, using magnetic field and magneto therapy can be more effective for patients who have a low percent
of stenosis and whose blood vessels or formed lumps on their
arterial walls are not yet rigid.
5. CONCLUSION
In this article, a transient study of magnetic field effect on a
stenosed artery, considering the arterial walls and atherosclerotic
plaque as a porous media, is investigated. The blood viscosity
treats as a non-Newtonian fluid and an infinite current plate
is fixed under the stenosis region to produce two vortices on
the lump’s edges. These vortices act as additional heat sources,
making the lump warmer and creating a suitable condition for
dissolving it. The applied external magnetic field increases shear
stress value on the stenosis region, which can rub out and wash
away the fatty deposited plaques. In addition, the simulation
results show that, for a magneto-therapy process, a lower porous
lump is more appropriate, since bigger vortices can create higher
temperature and penetration velocity along the target site.
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